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## G = C6.72+ 1+4order 192 = 26·3

### 7th non-split extension by C6 of 2+ 1+4 acting via 2+ 1+4/C2×D4=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C6 — C6.72+ 1+4
 Chief series C1 — C3 — C6 — C2×C6 — C2×Dic3 — C4×Dic3 — C23.26D6 — C6.72+ 1+4
 Lower central C3 — C2×C6 — C6.72+ 1+4
 Upper central C1 — C22 — C2×C4⋊C4

Generators and relations for C6.72+ 1+4
G = < a,b,c,d,e | a6=b4=1, c2=e2=a3, d2=b2, bab-1=dad-1=eae-1=a-1, ac=ca, cbc-1=b-1, dbd-1=a3b, be=eb, dcd-1=ece-1=a3c, ede-1=b2d >

Subgroups: 440 in 206 conjugacy classes, 111 normal (17 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C22×C4, C22×C4, C2×Q8, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, C2×C4⋊C4, C42⋊C2, C22⋊Q8, C42.C2, C4⋊Q8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×Dic6, C22×C12, C22×C12, C23.41C23, C12⋊Q8, C4.Dic6, C12.48D4, C23.26D6, C6×C4⋊C4, C6.72+ 1+4
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, C24, Dic6, C22×S3, C22×Q8, 2+ 1+4, 2- 1+4, C2×Dic6, S3×C23, C23.41C23, C22×Dic6, D46D6, Q8.15D6, C6.72+ 1+4

Smallest permutation representation of C6.72+ 1+4
On 96 points
Generators in S96
```(1 2 3 4 5 6)(7 8 9 10 11 12)(13 14 15 16 17 18)(19 20 21 22 23 24)(25 26 27 28 29 30)(31 32 33 34 35 36)(37 38 39 40 41 42)(43 44 45 46 47 48)(49 50 51 52 53 54)(55 56 57 58 59 60)(61 62 63 64 65 66)(67 68 69 70 71 72)(73 74 75 76 77 78)(79 80 81 82 83 84)(85 86 87 88 89 90)(91 92 93 94 95 96)
(1 93 14 7)(2 92 15 12)(3 91 16 11)(4 96 17 10)(5 95 18 9)(6 94 13 8)(19 83 27 90)(20 82 28 89)(21 81 29 88)(22 80 30 87)(23 79 25 86)(24 84 26 85)(31 68 39 75)(32 67 40 74)(33 72 41 73)(34 71 42 78)(35 70 37 77)(36 69 38 76)(43 59 52 66)(44 58 53 65)(45 57 54 64)(46 56 49 63)(47 55 50 62)(48 60 51 61)
(1 21 4 24)(2 22 5 19)(3 23 6 20)(7 81 10 84)(8 82 11 79)(9 83 12 80)(13 28 16 25)(14 29 17 26)(15 30 18 27)(31 52 34 49)(32 53 35 50)(33 54 36 51)(37 47 40 44)(38 48 41 45)(39 43 42 46)(55 67 58 70)(56 68 59 71)(57 69 60 72)(61 73 64 76)(62 74 65 77)(63 75 66 78)(85 93 88 96)(86 94 89 91)(87 95 90 92)
(1 57 14 64)(2 56 15 63)(3 55 16 62)(4 60 17 61)(5 59 18 66)(6 58 13 65)(7 48 93 51)(8 47 94 50)(9 46 95 49)(10 45 96 54)(11 44 91 53)(12 43 92 52)(19 68 27 75)(20 67 28 74)(21 72 29 73)(22 71 30 78)(23 70 25 77)(24 69 26 76)(31 80 39 87)(32 79 40 86)(33 84 41 85)(34 83 42 90)(35 82 37 89)(36 81 38 88)
(1 69 4 72)(2 68 5 71)(3 67 6 70)(7 36 10 33)(8 35 11 32)(9 34 12 31)(13 77 16 74)(14 76 17 73)(15 75 18 78)(19 59 22 56)(20 58 23 55)(21 57 24 60)(25 62 28 65)(26 61 29 64)(27 66 30 63)(37 91 40 94)(38 96 41 93)(39 95 42 92)(43 87 46 90)(44 86 47 89)(45 85 48 88)(49 83 52 80)(50 82 53 79)(51 81 54 84)```

`G:=sub<Sym(96)| (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,93,14,7)(2,92,15,12)(3,91,16,11)(4,96,17,10)(5,95,18,9)(6,94,13,8)(19,83,27,90)(20,82,28,89)(21,81,29,88)(22,80,30,87)(23,79,25,86)(24,84,26,85)(31,68,39,75)(32,67,40,74)(33,72,41,73)(34,71,42,78)(35,70,37,77)(36,69,38,76)(43,59,52,66)(44,58,53,65)(45,57,54,64)(46,56,49,63)(47,55,50,62)(48,60,51,61), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,81,10,84)(8,82,11,79)(9,83,12,80)(13,28,16,25)(14,29,17,26)(15,30,18,27)(31,52,34,49)(32,53,35,50)(33,54,36,51)(37,47,40,44)(38,48,41,45)(39,43,42,46)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,73,64,76)(62,74,65,77)(63,75,66,78)(85,93,88,96)(86,94,89,91)(87,95,90,92), (1,57,14,64)(2,56,15,63)(3,55,16,62)(4,60,17,61)(5,59,18,66)(6,58,13,65)(7,48,93,51)(8,47,94,50)(9,46,95,49)(10,45,96,54)(11,44,91,53)(12,43,92,52)(19,68,27,75)(20,67,28,74)(21,72,29,73)(22,71,30,78)(23,70,25,77)(24,69,26,76)(31,80,39,87)(32,79,40,86)(33,84,41,85)(34,83,42,90)(35,82,37,89)(36,81,38,88), (1,69,4,72)(2,68,5,71)(3,67,6,70)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,77,16,74)(14,76,17,73)(15,75,18,78)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,62,28,65)(26,61,29,64)(27,66,30,63)(37,91,40,94)(38,96,41,93)(39,95,42,92)(43,87,46,90)(44,86,47,89)(45,85,48,88)(49,83,52,80)(50,82,53,79)(51,81,54,84)>;`

`G:=Group( (1,2,3,4,5,6)(7,8,9,10,11,12)(13,14,15,16,17,18)(19,20,21,22,23,24)(25,26,27,28,29,30)(31,32,33,34,35,36)(37,38,39,40,41,42)(43,44,45,46,47,48)(49,50,51,52,53,54)(55,56,57,58,59,60)(61,62,63,64,65,66)(67,68,69,70,71,72)(73,74,75,76,77,78)(79,80,81,82,83,84)(85,86,87,88,89,90)(91,92,93,94,95,96), (1,93,14,7)(2,92,15,12)(3,91,16,11)(4,96,17,10)(5,95,18,9)(6,94,13,8)(19,83,27,90)(20,82,28,89)(21,81,29,88)(22,80,30,87)(23,79,25,86)(24,84,26,85)(31,68,39,75)(32,67,40,74)(33,72,41,73)(34,71,42,78)(35,70,37,77)(36,69,38,76)(43,59,52,66)(44,58,53,65)(45,57,54,64)(46,56,49,63)(47,55,50,62)(48,60,51,61), (1,21,4,24)(2,22,5,19)(3,23,6,20)(7,81,10,84)(8,82,11,79)(9,83,12,80)(13,28,16,25)(14,29,17,26)(15,30,18,27)(31,52,34,49)(32,53,35,50)(33,54,36,51)(37,47,40,44)(38,48,41,45)(39,43,42,46)(55,67,58,70)(56,68,59,71)(57,69,60,72)(61,73,64,76)(62,74,65,77)(63,75,66,78)(85,93,88,96)(86,94,89,91)(87,95,90,92), (1,57,14,64)(2,56,15,63)(3,55,16,62)(4,60,17,61)(5,59,18,66)(6,58,13,65)(7,48,93,51)(8,47,94,50)(9,46,95,49)(10,45,96,54)(11,44,91,53)(12,43,92,52)(19,68,27,75)(20,67,28,74)(21,72,29,73)(22,71,30,78)(23,70,25,77)(24,69,26,76)(31,80,39,87)(32,79,40,86)(33,84,41,85)(34,83,42,90)(35,82,37,89)(36,81,38,88), (1,69,4,72)(2,68,5,71)(3,67,6,70)(7,36,10,33)(8,35,11,32)(9,34,12,31)(13,77,16,74)(14,76,17,73)(15,75,18,78)(19,59,22,56)(20,58,23,55)(21,57,24,60)(25,62,28,65)(26,61,29,64)(27,66,30,63)(37,91,40,94)(38,96,41,93)(39,95,42,92)(43,87,46,90)(44,86,47,89)(45,85,48,88)(49,83,52,80)(50,82,53,79)(51,81,54,84) );`

`G=PermutationGroup([[(1,2,3,4,5,6),(7,8,9,10,11,12),(13,14,15,16,17,18),(19,20,21,22,23,24),(25,26,27,28,29,30),(31,32,33,34,35,36),(37,38,39,40,41,42),(43,44,45,46,47,48),(49,50,51,52,53,54),(55,56,57,58,59,60),(61,62,63,64,65,66),(67,68,69,70,71,72),(73,74,75,76,77,78),(79,80,81,82,83,84),(85,86,87,88,89,90),(91,92,93,94,95,96)], [(1,93,14,7),(2,92,15,12),(3,91,16,11),(4,96,17,10),(5,95,18,9),(6,94,13,8),(19,83,27,90),(20,82,28,89),(21,81,29,88),(22,80,30,87),(23,79,25,86),(24,84,26,85),(31,68,39,75),(32,67,40,74),(33,72,41,73),(34,71,42,78),(35,70,37,77),(36,69,38,76),(43,59,52,66),(44,58,53,65),(45,57,54,64),(46,56,49,63),(47,55,50,62),(48,60,51,61)], [(1,21,4,24),(2,22,5,19),(3,23,6,20),(7,81,10,84),(8,82,11,79),(9,83,12,80),(13,28,16,25),(14,29,17,26),(15,30,18,27),(31,52,34,49),(32,53,35,50),(33,54,36,51),(37,47,40,44),(38,48,41,45),(39,43,42,46),(55,67,58,70),(56,68,59,71),(57,69,60,72),(61,73,64,76),(62,74,65,77),(63,75,66,78),(85,93,88,96),(86,94,89,91),(87,95,90,92)], [(1,57,14,64),(2,56,15,63),(3,55,16,62),(4,60,17,61),(5,59,18,66),(6,58,13,65),(7,48,93,51),(8,47,94,50),(9,46,95,49),(10,45,96,54),(11,44,91,53),(12,43,92,52),(19,68,27,75),(20,67,28,74),(21,72,29,73),(22,71,30,78),(23,70,25,77),(24,69,26,76),(31,80,39,87),(32,79,40,86),(33,84,41,85),(34,83,42,90),(35,82,37,89),(36,81,38,88)], [(1,69,4,72),(2,68,5,71),(3,67,6,70),(7,36,10,33),(8,35,11,32),(9,34,12,31),(13,77,16,74),(14,76,17,73),(15,75,18,78),(19,59,22,56),(20,58,23,55),(21,57,24,60),(25,62,28,65),(26,61,29,64),(27,66,30,63),(37,91,40,94),(38,96,41,93),(39,95,42,92),(43,87,46,90),(44,86,47,89),(45,85,48,88),(49,83,52,80),(50,82,53,79),(51,81,54,84)]])`

42 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 4I ··· 4P 6A ··· 6G 12A ··· 12L order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 4 ··· 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 12 ··· 12 2 ··· 2 4 ··· 4

42 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 4 type + + + + + + + - + + - + - image C1 C2 C2 C2 C2 C2 S3 Q8 D6 D6 Dic6 2+ 1+4 2- 1+4 D4⋊6D6 Q8.15D6 kernel C6.72+ 1+4 C12⋊Q8 C4.Dic6 C12.48D4 C23.26D6 C6×C4⋊C4 C2×C4⋊C4 C2×C12 C4⋊C4 C22×C4 C2×C4 C6 C6 C2 C2 # reps 1 4 4 4 2 1 1 4 4 3 8 1 1 2 2

Matrix representation of C6.72+ 1+4 in GL6(𝔽13)

 10 0 0 0 0 0 0 4 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 0 5 0 0 0 0 5 0 0 0 0 0 0 0 0 0 7 5 0 0 0 0 6 6 0 0 6 8 0 0 0 0 7 7 0 0
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 0 0 0 0 12 11 0 0 0 0 0 1
,
 0 12 0 0 0 0 1 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 1 0 0 0 0 0 0 1 0 0
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 0 0 12 11 0 0 0 0 0 1 0 0 12 11 0 0 0 0 0 1 0 0

`G:=sub<GL(6,GF(13))| [10,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,5,0,0,0,0,5,0,0,0,0,0,0,0,0,0,6,7,0,0,0,0,8,7,0,0,7,6,0,0,0,0,5,6,0,0],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,12,0,0,0,0,0,11,1],[0,1,0,0,0,0,12,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,12,0,0],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,12,0,0,0,0,0,11,1,0,0] >;`

C6.72+ 1+4 in GAP, Magma, Sage, TeX

`C_6._72_+^{1+4}`
`% in TeX`

`G:=Group("C6.7ES+(2,2)");`
`// GroupNames label`

`G:=SmallGroup(192,1059);`
`// by ID`

`G=gap.SmallGroup(192,1059);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,232,758,675,570,80,6278]);`
`// Polycyclic`

`G:=Group<a,b,c,d,e|a^6=b^4=1,c^2=e^2=a^3,d^2=b^2,b*a*b^-1=d*a*d^-1=e*a*e^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1,d*b*d^-1=a^3*b,b*e=e*b,d*c*d^-1=e*c*e^-1=a^3*c,e*d*e^-1=b^2*d>;`
`// generators/relations`

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